Membership algebra as a logical framework for equational specification | SpringerLink
Skip to main content
Springer Nature Link
Log in

Membership algebra as a logical framework for equational specification

  • Invited Papers
  • Conference paper
  • First Online:
Recent Trends in Algebraic Development Techniques (WADT 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1376))

Included in the following conference series:

Abstract

This paper proposes membership equational logic—a Horn logic in which the basic predicates are equations t = t′ and membership assertions t : s stating that a term t belongs to a sort s—as a logical framework in which a very wide range of total and partial equational specification formalisms can be naturally represented. Key features of this logic include: simplicity, liberality and equational character; generality and expressiveness in supporting subsorts, overloading, errors and partiality; and efficient implementability in systems such as Maude. The paper presents the basic properties of the logic and its models, and discusses in detail how many total and partial equational specification formalisms, including order-sorted algebra and partial membership equational logic, can be represented in it, as well as the practical benefits in terms of tool reusability that this opens up for other languages, including CASL.

Supported by Office of Naval Research Contracts N00014-95-C-0225 and N00014-96 C-0114, National Science Foundation Grant CCR-9633363, and by the Advanced Software Enrichment Project of the Information-Technology Promotion Agency, Japan (IPA).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. J. Adámek and J. Rosický. Locally Presentable and Accessible Categories. Cambridge University Press, 1994.

    Google Scholar 

  2. M. Barr and C. Wells. Toposes, Triples and Theories. Springer-Verlag, 1985.

    Google Scholar 

  3. A. Bouhoula, J.-P. Jouannaud, and J. Meseguer. Specification and proof in membership equational logic. Manuscript, SRI International, August 1996.

    Google Scholar 

  4. A. Bouhoula, J.-P. Jouannaud, and J. Meseguer. Specification and proof in membership equational logic. In M. Bidoit and M. Dauchet, editors, Proceedings TAPSOFT'97, volume 1214 of Lecture Notes in Computer Science. Springer-Verlag, 1997.

    Google Scholar 

  5. P. Burmeister. Partial algebras-Survey of a unifying approach towards a twovalued model theory for partial algebras. Algebra Universalis, 15:306–358, 1982.

    Google Scholar 

  6. M. Cerioli and J. Meseguer. May I borrow your logic? (Transporting logical structure along maps). Theoretical Computer Science, 173:311–347, 1997.

    Google Scholar 

  7. M. G. Clavel, S. Eker, P. Lincoln, and J. Meseguer. Principles of Maude. In J. Meseguer, editor, Proc. First Intl. Workshop on Rewriting Logic and its Applications, volume 4 of Electronic Notes in Theoretical Computer Science. Elsevier, 1996. http://wwwi.elsevier.nl/mcs/tcs/pc/volume4.htm.

    Google Scholar 

  8. M. G. Clavel and J. Meseguer. Reflection and strategies in rewriting logic. In J. Meseguer, editor, Proc. First Intl. Workshop on Rewriting Logic and its Applications, volume 4 of Electronic Notes in Theoretical Computer Science. Elsevier, 1996. http://wwwi.elsevier.nl/mcs/tcs/pc/volume4.htm.

    Google Scholar 

  9. CoFI Task Group on Semantics. CASL—The CoFI algebraic specification language, version 0.97, Semantics. http://www.brics.dk/Projects/CoFI, July 1997.

    Google Scholar 

  10. M. Coste. Localisation, spectra and sheaf representation. In M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors, Applications of Sheaves, volume 753 of Lecture Notes in Mathematics, pages 212–238. Springer-Verlag, 1979.

    Google Scholar 

  11. P. Freyd. Aspects of topoi. Bull. Austral. Math. Soc., 7:1–76, 1972.

    Google Scholar 

  12. P. Gabriel and F. Ulmer. Lokal prdsentierbare Kategorien. Springer Lecture Notes in Mathematics No. 221, 1971.

    Google Scholar 

  13. J. Goguen and R. Burstall. Institutions: Abstract model theory for specification and programming. Journal of the ACM, 39(1):95–146, 1992.

    Article  Google Scholar 

  14. J. Goguen, J.-P. Jouannaud, and J. Meseguer. Operational semantics of ordersorted algebra. In W. Brauer, editor, Proceedings, 1985 International Conference on Automata, Languages and Programming, volume 194 of Lecture Notes in Computer Science, pages 221–231. Springer-Verlag, 1985.

    Google Scholar 

  15. J. Goguen and J. Meseguer. Models and equality for logical programming. In H. Ehrig, G. Levi, R. Kowalski, and U. Montanari, editors, Proceedings TAPSOFT'87 volume 250 of Lecture Notes in Computer Science, pages 1–22. Springer-Verlag, 1987.

    Google Scholar 

  16. J. Goguen and J. Meseguer. Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science, 105:217–273, 1992.

    Google Scholar 

  17. C. Hintermeier, C. Kirchner, and H. Kirchner. Dynamically-typed computations for order-sorted equational presentations. In Proc. ICALP'94, pages 60–75. Springer LNCS 510, 1994.

    Google Scholar 

  18. C. Hintermeier, C. Kirchner, and H. Kirchner. Dynamically-typed computations for order-sorted equational presentations. Technical Report 2208, INRIA, March 1994.

    Google Scholar 

  19. H.-J. Kreowski and T. Mossakowski. Equivalence and difference between institutions: simulating Horn Clause Logic with based algebras. Math. Struct. in Comp. Sci., 5:189–215, 1995.

    Google Scholar 

  20. S. MacLane. Categories for the Working Mathematician. Springer-Verlag, 1971.

    Google Scholar 

  21. V. Manca, A. Salibra, and G. Scollo. Equational type logic. Theoretical Computer Science, 77:131–159, 1990.

    Google Scholar 

  22. N. Martí-Oliet and J. Meseguer. Rewriting logic as a logical and semantic framework. In J. Meseguer, editor, Proc. First Intl. Workshop on Rewriting Logic and its Applications, volume 4 of Electronic Notes in Theoretical Computer Science. Elsevier, 1996. http://wwwl.elsevier.nl/mcs/tcs/pc/volume4.htm.

    Google Scholar 

  23. J. Meseguer. Membership algebra. Lecture and abstract at the Dagstuhl Seminar on “Specification and Semantics,” July 9, 1996.

    Google Scholar 

  24. J. Meseguer. General logics. In H.-D. E. et al., editor, Logic Colloquium'87, pages 275–329. North-Holland, 1989.

    Google Scholar 

  25. J. Meseguer and J. Goguen. Order-sorted algebra solves the constructor-selector, multiple representation and coercion problems. Information and Computation, 103(1):114–158, 1993.

    Google Scholar 

  26. J. Meseguer and U. Montanari. Mapping tile logic into rewriting logic. In F. Parisi-Presicce, ed., Proc. WADT'97, Springer LNCS, this volume, 1998.

    Google Scholar 

  27. T. Mossakowski. Equivalences among various logical frameworks of partial algebras. In H. K. Büning, editor, Computer Science Logic, Paderborn, Germany, September 1995, Selected Papers, volume 1092 of Lecture Notes in Computer Science, pages 403–433. Springer-Verlag, 1996.

    Google Scholar 

  28. P. D. Mosses. Unified algebras and institutions. In Proc. Fourth Annual IEEE Symp. on Logic in Computer Science, pages 304–312, Asilomar, California, June 1989.

    Google Scholar 

  29. P. D. Mosses. The use of sorts in algebraic specifications. In M. Bidoit and C. Choppy, editors, Recent trends in Data Type Specification, 8th WADT, August 1991, volume 655 of Lecture Notes in Computer Science, pages 66–91. Springer-Verlag, 1993.

    Google Scholar 

  30. F. Parisi-Presicce and S. Veglioni. Heterogeneous unified algebras. In Proceedings of MFCS'93, 18th International Symposium on Mathematical Foundations of Computer Science, pages 618–628. Springer LNCS 711, 1993.

    Google Scholar 

  31. A. Poigné. Parametrization for order-sorted algebraic specification. Journal of Computer and System Sciences, 40(2):229–268, 1990.

    Google Scholar 

  32. H. Reichel. Initial Computability, Algebraic Specifications, and Partial Algebras. Oxford University Press, 1987.

    Google Scholar 

  33. G. F. Stuart and W. W. Wadge. Classified model abstract data type specification. Manuscript, University of Victoria, 1991.

    Google Scholar 

  34. W. W. Wadge. Classified algebras. Technical report, University of Warwick, 1982.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SRI International, 94025, Menlo Park, CA, USA

    José Meseguer

Authors
  1. José Meseguer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Francesco Parisi Presicce

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Meseguer, J. (1998). Membership algebra as a logical framework for equational specification. In: Presicce, F.P. (eds) Recent Trends in Algebraic Development Techniques. WADT 1997. Lecture Notes in Computer Science, vol 1376. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-64299-4_26

Download citation

  • DOI: https://doi.org/10.1007/3-540-64299-4_26

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64299-2

  • Online ISBN: 978-3-540-69719-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation